- Email: [email protected]
Effective hybridization in mixed valence systems
302
Journal of Magnetism
EFFECTIVE
HYBRIDIZATION
M.D. NUNEZ-REGUEIRO L.uhoratoire d’Etudes des Prop&v&
IN MIXED VALENCE
and Magnetic
Materials 478~48 (1985) 302-304 North-Holland. Amsterdam
SYSTEMS
and M. AVIGNON Electrontques
da Soltdes, CNRS.
B. P. 160. 38042 Grenoble Cede-r, Frunce
Within the framework of the Anderson model. we present a systematic study of the renormalization of the hyhridlzation by the f-d Coulomb repulsion and by the electron-phonon coupling. Their influence on the valence transition is studied. The possibility of a phonon induced on-site hybridization is discussed.
1. Introduction
H,=~ELd;dL+~EOf,?f,+~(V~f:d~+h.c.).(l) h
In spite of the experimental
indications concerning the existence of so-called ‘hybridization gaps’ in some IV compounds (Sm$ [l], TmSe [2] ), their origin remains controversial. This question is related to the existence of an on-site (k-independent) hybridization V between localized f and itinerant d states, in principle forbidden by symmetry, but however necessary to open the gap. Intersite hybridization being k-dependent generally does not open a gap. Compared to constant hybridization, the k-dependence of hybridization does not introduce important differences in the change of valence under variation of electronic parameters 131. On the other hand, several mechanisms are known to induce indirect hybridization with intra-atomic character such as the intra-atomic Coulomb interaction between d and f electrons (Falicov model) [4] or local electron-phonon couplings [5,6]. This renormalizes the direct hybridization term. The character of the renormalization becomes different depending on whether direct hybridization is considered constant or k-dependent.
I
H,,=xG H,,=
d:d,f,+f,.
We thus reconsider the spinless periodic Anderson model (HA) within the Hartree-Fock approximation to perform a systematic study of the hybridization and its renormalization fi by the f-d Coulomb repulsion G (H,,) and by the electron-phonon interaction (H,,; the usual f electron-phonon coupling g, and the phonon induced 4f-5d transitions are considered). 0304-8853/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
+h)
It is shown that a constant direct V is renormalized in a similar form by these two interactions H,, and H,, and that contrary to previous results the valence transition is not related to a change of sign of p [7] nor to its cancellation [6]. The nature of the transition is studied as a function of the different coupling parameters and the possibility of inducing on-site hybridization is diacussed. Assuming then an on-site hybridization, its renormalization by the two terms (2) and (3) within the Hartree-Fock approximation is given by: v=
V-
Gu
for H,,.
(4)
p=V-a(l-n,)-2G,u
forH,,.
Then for a constant &-Iln
W
A+
(5)
and
i ’ h \k: + 4li’j’
fz=(f,+d,)=-;1;x
hybridization
(2)
-C[g,nl+g,(f:d,+h.c.>](h,+
where G, = 2g,‘/Aw 2. Effective
I,
density
w+\i”[i+
of states of bandwidth
W:
w]‘+41F’1* (7)
d’+
l”d*+4/PI2
A similar expression has been obtained to first order by a renormalization group method by Schlottmann (81
M.D. Nuner Regueiro, M. Avignon / Effective hybridization in mrxed valence systems
Fig. 1. /( p) = 6’(1+ era) for a = 0.5, (a) d = 0.5 or - 1.5, (c) d = 0 or - 1, (d) d = - 0.5 in units of W.
Fig. 2. Average number of conduction electrons and renormalized hybridization as a function of the position of the f-level for G,/V=3,
for the Falicov-Kimball model (Hrk); although in that case, it does not show the dependence on i\, the relative position of the band and the f-level. His expression thus corresponds to d being set equal to zero and also to the limit W % I? In the case of phonons, Giner and Brouers 17) proposed a second-order equation to obtain the renormalized fi, but it must be noted that their coefficient AU is also v dependent, thus giving formally a more complicate self-consistent equation. In fig. 1, we plot /(v) = p(1 + au), (Y= G or 2G,. With a semi-elliptic density of states of the same bandwidth, we have obtained a very similar set of curves with only minor differences. This gives a more general character to these curves. Note the change in the behaviour of f(v) as the localized level E, enters the conduction band (decreasing d). The intersection of f(P) with V yields the renormalized p in the H,, case. For small ]d( = /A, + G(1 - 2nJ one or three solutions appear depending on the V value, and it is always the solution with greatest p which has the lowest energy. The same procedure can be used for phonons, H,,, looking for the intersection of f(p) with V - m (1 - n,). In this case, p can also cancel out or change sign. The band energy is not modified but the renormalization of the f-level: Lo = E, - G, (1 - n,) - 2m~ implies a self-consistent loop in the determination of J?
3. Valence transitions 1) Falicov-Kimball model: the solution with greatest p yields a smooth valence transition. The two other solutions give a loop with higher energy. The discontinuous transitions for G > G,,cr” = /m found, when excitonic correlations are neglected, disappear with the renormalized I? The same result is obtained in the atomic limit: the excitonic contribution
303
G,/V=0.2
and V/W=l.
exactly compensates the term which gives the discontinuity in the former case. 2) Case G, = 0. V is unchanged and as in the H,, case, neglecting excitonic terms, discontinuous transitions are found for G, z Gf”’ = 2GFi. Entel et al. [6] obtained a smooth transition because their G, < Gy”. 3) Case G, = 0. V is renormalized but not the electron energies. As in l), only continuous transitions are found, the other p solutions yield a loop. 4) Case G, and G, # 0. V and E, are renormalized. The critical value Gy”’ to have a discontinuous transition in 2) is lowered when G, # 0. For the parameter values of Entel et al. [6], we also found that p cancels at the valence transition but fig. 2 shows that this is not a necessary condition to have the discontinuity. If V > m for enough large G, and G, values, there is a jumplike transition for p always positive. The problem is if for such large values of the parameters, one can still use the Hartree-Fock approximation. Introducing a phenomenological dependence of the parameters on the volume to simulate the effect of pressure, Izmailyan et al. [9] have also found the possibility of a discontinuous transition for Per, + 0. The same results are obtained analytically in the atomic limit.
4. Discussion The general form of the excitonic contribution to the renormalization of the hybridization has been analysed within the Hartree-Fock approximation. In contrast to previous results, it has been found that an felectron-phonon coupling is sufficient to produce a discontinuous transition, for G, > Gy”. This critical value is considerably lowered if phonon induced 4f-5d transitions are present. In that case, and in contrast to
304
M.D. Nunez
Regueiro, h4. Aurgmn
/ Effective hyhriduation
VI mxed
the H,, model, the renormalized v can cancel out or change sign, but none of these possibilities is necessary
should
to have
the hybridization
can
a discontinuity.
occur
for
A jumplike
v keeping
the same
valence
transition
us
mention
here
,
hybridization.
be explored
further. is necessary
These
different
Still better to settle
possibilities
understanding
of
the issue.
that
I’(k) P(k)=V(k)+a(
on-site
sign.
if the k-dependence of hybridization is included, then eqs. (4)-(6) give rise to an integral equation for P(k); P(k) contains two parts: a k-dependent one V(k) and a constant on-site one. In general, such an equation cannot be solved, except numerically. In a first-order approximation: Let
direct
valence systems
_
d3k
(8)
The phonon contribution to the on-site hybridization requires local d ++ f electron-phonon processes (g, f 0). Note that such processes may exist even in the absence of direct on-site hybridization. Due to their difference in character, d and f wave functions should react differently to ionic displacements. Localized f electrons will respond as core electrons, following rigidly the ions, while extended d wave functions will not. During lattice vibrations the local symmetry is broken and on-site hybridization possible. This approach is quite different from the mechanism invoked recently by Alascio et al. [lo]. They proposed that a lattice distortion breaking the symmetry of the conduction states may produce a
References 111 S. Von Molnar, T. Theis, A. Benoit, A. Briggs, J. Flouquet,
PI
[31 [41 [51 [61 [71 PI (91 PO1
R. Ravex and 2. Fisk, Valence Instabilities, eds. P. Wachter and H. Boppart (North-Holland, Amsterdam, 1982) p. 389. F. Lapierre. M. Mignot, J. Flouquet, P. Haen, M. Ribault and F. Holtzberg, Valence Fluctuations in Solids, eds. L.M. Falicov. W. Hanke and M.B. Maple (North-Holland. Amsterdam, 1981) p. 305. M. Avignon and G. Spronken. (to be published). 0.1. Khomskii and A.N. Kocharyan, Solid State Commun. 18 (1976) 985. M. Avignon, F. Brouers and K.H. Bennemann, J. de Phys. Colloq. 40 (1979) c5, 377. P. Entel, H.J. Leder and N. Grewe. Z. Phys. B 30 (1978) 277. J. Giner and F. Brouers, Phys. Rev. B25 (1982) 5214. P. Schlottmann, Phys. Rev. B22 (1980) 613. NSh. Izmailyan, A.N. Kocharyan, P.S. Ovnanyan and D.I. Khomskii, Sov. Phys. Solid State 23 (1981) 2977. B.R. Alascio, A.A. Aligia and C.A. Balseiro, Solid State Commun. 48 (1983) 185.
Journal of Magnetism
EFFECTIVE
HYBRIDIZATION
M.D. NUNEZ-REGUEIRO L.uhoratoire d’Etudes des Prop&v&
IN MIXED VALENCE
and Magnetic
Materials 478~48 (1985) 302-304 North-Holland. Amsterdam
SYSTEMS
and M. AVIGNON Electrontques
da Soltdes, CNRS.
B. P. 160. 38042 Grenoble Cede-r, Frunce
Within the framework of the Anderson model. we present a systematic study of the renormalization of the hyhridlzation by the f-d Coulomb repulsion and by the electron-phonon coupling. Their influence on the valence transition is studied. The possibility of a phonon induced on-site hybridization is discussed.
1. Introduction
H,=~ELd;dL+~EOf,?f,+~(V~f:d~+h.c.).(l) h
In spite of the experimental
indications concerning the existence of so-called ‘hybridization gaps’ in some IV compounds (Sm$ [l], TmSe [2] ), their origin remains controversial. This question is related to the existence of an on-site (k-independent) hybridization V between localized f and itinerant d states, in principle forbidden by symmetry, but however necessary to open the gap. Intersite hybridization being k-dependent generally does not open a gap. Compared to constant hybridization, the k-dependence of hybridization does not introduce important differences in the change of valence under variation of electronic parameters 131. On the other hand, several mechanisms are known to induce indirect hybridization with intra-atomic character such as the intra-atomic Coulomb interaction between d and f electrons (Falicov model) [4] or local electron-phonon couplings [5,6]. This renormalizes the direct hybridization term. The character of the renormalization becomes different depending on whether direct hybridization is considered constant or k-dependent.
I
H,,=xG H,,=
d:d,f,+f,.
We thus reconsider the spinless periodic Anderson model (HA) within the Hartree-Fock approximation to perform a systematic study of the hybridization and its renormalization fi by the f-d Coulomb repulsion G (H,,) and by the electron-phonon interaction (H,,; the usual f electron-phonon coupling g, and the phonon induced 4f-5d transitions are considered). 0304-8853/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
+h)
It is shown that a constant direct V is renormalized in a similar form by these two interactions H,, and H,, and that contrary to previous results the valence transition is not related to a change of sign of p [7] nor to its cancellation [6]. The nature of the transition is studied as a function of the different coupling parameters and the possibility of inducing on-site hybridization is diacussed. Assuming then an on-site hybridization, its renormalization by the two terms (2) and (3) within the Hartree-Fock approximation is given by: v=
V-
Gu
for H,,.
(4)
p=V-a(l-n,)-2G,u
forH,,.
Then for a constant &-Iln
W
A+
(5)
and
i ’ h \k: + 4li’j’
fz=(f,+d,)=-;1;x
hybridization
(2)
-C[g,nl+g,(f:d,+h.c.>](h,+
where G, = 2g,‘/Aw 2. Effective
I,
density
w+\i”[i+
of states of bandwidth
W:
w]‘+41F’1* (7)
d’+
l”d*+4/PI2
A similar expression has been obtained to first order by a renormalization group method by Schlottmann (81
M.D. Nuner Regueiro, M. Avignon / Effective hybridization in mrxed valence systems
Fig. 1. /( p) = 6’(1+ era) for a = 0.5, (a) d = 0.5 or - 1.5, (c) d = 0 or - 1, (d) d = - 0.5 in units of W.
Fig. 2. Average number of conduction electrons and renormalized hybridization as a function of the position of the f-level for G,/V=3,
for the Falicov-Kimball model (Hrk); although in that case, it does not show the dependence on i\, the relative position of the band and the f-level. His expression thus corresponds to d being set equal to zero and also to the limit W % I? In the case of phonons, Giner and Brouers 17) proposed a second-order equation to obtain the renormalized fi, but it must be noted that their coefficient AU is also v dependent, thus giving formally a more complicate self-consistent equation. In fig. 1, we plot /(v) = p(1 + au), (Y= G or 2G,. With a semi-elliptic density of states of the same bandwidth, we have obtained a very similar set of curves with only minor differences. This gives a more general character to these curves. Note the change in the behaviour of f(v) as the localized level E, enters the conduction band (decreasing d). The intersection of f(P) with V yields the renormalized p in the H,, case. For small ]d( = /A, + G(1 - 2nJ one or three solutions appear depending on the V value, and it is always the solution with greatest p which has the lowest energy. The same procedure can be used for phonons, H,,, looking for the intersection of f(p) with V - m (1 - n,). In this case, p can also cancel out or change sign. The band energy is not modified but the renormalization of the f-level: Lo = E, - G, (1 - n,) - 2m~ implies a self-consistent loop in the determination of J?
3. Valence transitions 1) Falicov-Kimball model: the solution with greatest p yields a smooth valence transition. The two other solutions give a loop with higher energy. The discontinuous transitions for G > G,,cr” = /m found, when excitonic correlations are neglected, disappear with the renormalized I? The same result is obtained in the atomic limit: the excitonic contribution
303
G,/V=0.2
and V/W=l.
exactly compensates the term which gives the discontinuity in the former case. 2) Case G, = 0. V is unchanged and as in the H,, case, neglecting excitonic terms, discontinuous transitions are found for G, z Gf”’ = 2GFi. Entel et al. [6] obtained a smooth transition because their G, < Gy”. 3) Case G, = 0. V is renormalized but not the electron energies. As in l), only continuous transitions are found, the other p solutions yield a loop. 4) Case G, and G, # 0. V and E, are renormalized. The critical value Gy”’ to have a discontinuous transition in 2) is lowered when G, # 0. For the parameter values of Entel et al. [6], we also found that p cancels at the valence transition but fig. 2 shows that this is not a necessary condition to have the discontinuity. If V > m for enough large G, and G, values, there is a jumplike transition for p always positive. The problem is if for such large values of the parameters, one can still use the Hartree-Fock approximation. Introducing a phenomenological dependence of the parameters on the volume to simulate the effect of pressure, Izmailyan et al. [9] have also found the possibility of a discontinuous transition for Per, + 0. The same results are obtained analytically in the atomic limit.
4. Discussion The general form of the excitonic contribution to the renormalization of the hybridization has been analysed within the Hartree-Fock approximation. In contrast to previous results, it has been found that an felectron-phonon coupling is sufficient to produce a discontinuous transition, for G, > Gy”. This critical value is considerably lowered if phonon induced 4f-5d transitions are present. In that case, and in contrast to
304
M.D. Nunez
Regueiro, h4. Aurgmn
/ Effective hyhriduation
VI mxed
the H,, model, the renormalized v can cancel out or change sign, but none of these possibilities is necessary
should
to have
the hybridization
can
a discontinuity.
occur
for
A jumplike
v keeping
the same
valence
transition
us
mention
here
,
hybridization.
be explored
further. is necessary
These
different
Still better to settle
possibilities
understanding
of
the issue.
that
I’(k) P(k)=V(k)+a(
on-site
sign.
if the k-dependence of hybridization is included, then eqs. (4)-(6) give rise to an integral equation for P(k); P(k) contains two parts: a k-dependent one V(k) and a constant on-site one. In general, such an equation cannot be solved, except numerically. In a first-order approximation: Let
direct
valence systems
_
d3k
(8)
The phonon contribution to the on-site hybridization requires local d ++ f electron-phonon processes (g, f 0). Note that such processes may exist even in the absence of direct on-site hybridization. Due to their difference in character, d and f wave functions should react differently to ionic displacements. Localized f electrons will respond as core electrons, following rigidly the ions, while extended d wave functions will not. During lattice vibrations the local symmetry is broken and on-site hybridization possible. This approach is quite different from the mechanism invoked recently by Alascio et al. [lo]. They proposed that a lattice distortion breaking the symmetry of the conduction states may produce a
References 111 S. Von Molnar, T. Theis, A. Benoit, A. Briggs, J. Flouquet,
PI
[31 [41 [51 [61 [71 PI (91 PO1
R. Ravex and 2. Fisk, Valence Instabilities, eds. P. Wachter and H. Boppart (North-Holland, Amsterdam, 1982) p. 389. F. Lapierre. M. Mignot, J. Flouquet, P. Haen, M. Ribault and F. Holtzberg, Valence Fluctuations in Solids, eds. L.M. Falicov. W. Hanke and M.B. Maple (North-Holland. Amsterdam, 1981) p. 305. M. Avignon and G. Spronken. (to be published). 0.1. Khomskii and A.N. Kocharyan, Solid State Commun. 18 (1976) 985. M. Avignon, F. Brouers and K.H. Bennemann, J. de Phys. Colloq. 40 (1979) c5, 377. P. Entel, H.J. Leder and N. Grewe. Z. Phys. B 30 (1978) 277. J. Giner and F. Brouers, Phys. Rev. B25 (1982) 5214. P. Schlottmann, Phys. Rev. B22 (1980) 613. NSh. Izmailyan, A.N. Kocharyan, P.S. Ovnanyan and D.I. Khomskii, Sov. Phys. Solid State 23 (1981) 2977. B.R. Alascio, A.A. Aligia and C.A. Balseiro, Solid State Commun. 48 (1983) 185.